Sequence Of Knots Research Articles

Trajectory Planning and Speed Control for a Two-Link Rigid Manipulator

Trajectory Planning and Speed Control for a Two-Link Rigid Manipulator

PLANNING TRAJECTORY WITH SPEED CONTROLLED MANOEUVRES FOR A TWO-LINK RIGID MANIPULATOR

A two part trajectory and speed planning procedure for a two-link rigid manipulator is presented. The planning is done at the joint level using cubic spline functions, and the trajectory of the robot is specified by a sequence of knots in space Cartesian coordinates. These knots are transformed into two sets of joint coordinates, and piecewise cubic spline functions are used to fit these two sets employing a time scale to construct an initial trajectory for the manipulator. Linear scaling of the time variable is used to accommodate the angular velocity and acceleration constraints imposed at the joint level. A new nonlinear time scaling scheme is then used for speed control so as to fit the manipulator's tip velocity to a pre-specified profile. Simulation results for the manipulator with payload masses following a planned trajectory are presented.

Vassiliev invariants and knot polynomials

We give a criterion to detect whether the derivatives of knot polynomials at a point are Vassiliev invariants or not. As an application we show that for each nonnegative integer n, JK(n)(a) is a Vassiliev invariant if and only if a=1, where JK(n)(a) is the nth derivative of the Jones polynomial JK(t) of a knot K at t=a. Similarly we apply the criterion for the Conway, Alexander, Q-, HOMFLY and Kauffman polynomial.Also we give two methods of constructing a polynomial invariant from a numerical Vassiliev invariant of degree n, by using a sequence of knots induced from a double dating tangle. These two new polynomial invariants are Vassiliev invariants of degree ⩽n and their values on a knot are also polynomials of degree ⩽n.

Les invariants rationnels de type fini ne distinguent pas les nœuds dans SS2× SS1

We introduce geometric sequences of knots and establish the following criterion: if v is a rational invariant of degree ≤ m in the sense of Vassiliev, then v is a polynomial of degree ≤ m on every geometric sequence of knots. The torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: we construct knots in SS 2× SS 1 which cannot be distinguished by rational invariants of finite type. They can, however, be distinguished by invariants of finite type with values in a finite abelian group.

TWIST SEQUENCES AND VASSILIEV INVARIANTS

In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.

REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND THEIR INTERPRETATION AS MOVES TO MOVIES

A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. In the cross sections, an immersed collection of curves appears, and these are lifted to knot diagrams by using the projection direction from 4-space. We give a set of 15 moves to movies (called movie moves) such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This result generalizes the Roseman moves which are moves on projections where a height function has not been specified. The first 7 of the movie moves are height function parametrized versions of those given by Roseman. The remaining 8 are moves in which the topology of the projection remains unchanged.

Triangular-patch spline surfaces with symmetric labels

B-spline curve segments can be defined using a de Casteljau approach based on symmetric functions and a labelling of the control points in terms of subsequences taken from the full sequence of knots. These ideas are extended, and a new family of triangular-patch spline surfaces is proposed. The defining symmetric functions and the labelling for the control points are discussed.

Symmetric recursive algorithms for surfaces: B-patches and the de boor algorithm for polynomials over triangles

Using the concept of a symmetric recursive algorithm, we construct a new patch representation for bivariate polynomials: the B-patch. B-patches share many properties with B-spline segments: they are characterized by their control points and by a three-parameter family of knots. If the knots in each family coincide, we obtain the Bezier representation of a bivariate polynomial over a triangle. Therefore B-patches are a generalization of Bezier patches. B-patches have a de Boor-like evaluation algorithm, and, as in the case of B-spline curves, the control points of a B-patch can be expressed by simply inserting a sequence of knots into the corresponding polar form. In particular, this implies linear independence of the blending functions. B-patches can be joined smoothly and they have an algorithm for knot insertion that is completely similar to Boehm's algorithm for curves.